There is one major area of difference between these two schools: their governance.
St Paul's remains in the hands of the Mercers' Company, the self styled "premier" City Livery Company, whose members come largely from a small number of landed English families. The Company runs what it calls a "portfolio" of schools, including SPGS, Abingdon and the state funded Hammersmith Academy.
Governorships of the "portfolio" schools rotate amongst these families and down the generations. Lord Lucas, owner and editor of the Good Schools Guide, is a Liveryman in the Mercers and is the brother of the Hon Timothy Palmer, former Master of the Company and current Chair of Governors of SPGS.
Whether a club for landed gentry is an appropriate organisation to run schools in the 21st century is open to question. The trustees of St Paul's School are currently under investigation by the Charity Commission following the reports of child abuse and allegations of serial cover ups which emerged last year:
www.gov.uk/government/news/charity-commission-announces-investigation-into-st-pauls-school
The governing body of Westminster School is a good deal more diverse. Its link to the Abbey is historic but I think it would generally be agreed that the practices and outlook of the Church of England are more modern and transparent than those of the livery companies.
It is unfortunate that the livery companies were not abolished when the opportunity arose at the time of the Royal Commission into their fitness for purpose in 1884. SPS could then have been handed back into the care of the Church.
Parents too rarely look into the composition of boards of governors. The headmaster of St Paul's is an immensely talented and thoroughly decent man, by all accounts. The management of the school under his leadership should be unimpeachable. However, questions can legitimately be raised, and are being raised by the Charity Commission, about the quality of the school's governance.